The structure of the critical set in the mountain-pass theorem for nondifferentiable functions.
The paper studies the structure of the critical set at the level given by the minimax value in the mountain pass theorem for nonsmooth functionals expressed as a sum of a locally Lipschitz function and a proper, convex, lower semicontinuous function on a Banach space. The analysis relies on suitable deformation properties which hold for this class of nonsmooth functionals. One obtains significant information on the nature of the critical points, for instance, establishing the presence of a saddle-point. The authors' theorems extend the classical results due to [\textit{P. Pucci} and \textit{J. Serrin}, ``Extensions of the mountain pass theorem'', J. Funct. Anal. 59, 185--210 (1984; Zbl 0564.58012)]. A special attention is paid in the final part of the paper to the case of purely locally Lipschitz functionals.
scientific article; zbMATH DE number 2004937